1. Field
This invention relates to a method for determining parameters of a material. In particular this invention relates to the use of x-ray scattering to determine the parameters of a material. The material may, for example, be a semiconductor sample that consists of a number of very thin layers. The parameters of particular interest are the composition and thickness of these layers.
2. Description of the Related Art
It is well known to analyse a material sample by submitting it to x-ray analysis to produce an x-ray scattering profile. When analysing a semiconductor substrate the actual profile is compared to an expected profile for an approximation of the material. If the two profiles are different, then the expected profile is modified on the basis of error data generated from comparing the two profiles. This process is then repeated until the two profiles substantially match, at which point the parameters of the material under analysis are known from the parameters used for the last modified expected profile.
This technique is satisfactory for a large number of applications, but has certain weaknesses. Fitting predicted models to experimental data can be complex especially when physical mechanism giving rise to the profile is very sensitive to the wanted parameters. X-ray scattering procedures fall into this category and require model modification and comparisons in an iterative procedure to achieve the parameters of interest. The existence of several minima in the discrepancies between the calculated and experimental profiles often leads to incorrect solutions in an automatic refinement process. A number of methods of improving this process have been proposed.
U.S. Pat. No. 5,442,676 relates to a method of determining a given characteristic of a material sample. Measurements are made on a sample to obtain an experimental profile having structural features determined at least in part by the given characteristic and an expected profile is calculated for the sample using selected parameters. A degree of smoothing is applied to the experimental profile to reduce the structural features thereby producing a smoothed experimental profile and the same degree of smoothing is applied to the calculated profile to produce a smoothed calculated profile. The smoothed calculated profile is compared with the smoothed experimental profile to determine the difference between the smoothed profiles. The calculated profile is then modified by varying at least one of the parameters until the smoothed modified profile fits the smoothed experimental profile. The above steps are then repeated with the modified calculated profile using each time a degree of smoothing less than the previous time so that the structural features return and the final modified calculated profile provides a desired fit to the experimental profile thereby enabling the given characteristic to be determined from the parameters used for the final modified profile. This should generally reduce the computation time required for the fitting procedure, especially if the initial guess is not close. In addition, the smoothing of the experimental and calculated profiles removes or at least reduces the possibility of satellite or false minima occurring in the fitting procedure which would otherwise increase the possibility of a false result or at least cause an unnecessary increase in computation time.
In the paper xe2x80x9cApplication of genetic algorithms for characterization of thin layered materials by glancing incidence x-ray reflectometryxe2x80x9d by A. D. Dane et al., published 30 Jan. 1998 in the journal Physica B, the authors propose the use of known genetic algorithms for the characterisation of materials. The genetic algorithm is an optimisation technique and is used during the process of comparing the two x-ray profiles and modifying of the calculated profile. The genetic algorithm is able to find good fits within a single run. This reduces the amount of human effort and expertise required for analysing reflectivity measurements. Furthermore, it reduces the probability of overlooking feasible solutions.
U.S. Pat. No. 5,530,732 discloses a method of determining the composition and thicknesses of metamorphic layers at heterointerfaces of periodic laminated structures, such as multiple quantum well structures. An x-ray diffraction pattern of the actual structure is measured and a theoretical x-ray diffraction pattern is calculated using dynamic x-ray theory and giving special attention to x-ray diffraction fringes near a satellite peak in the pattern. The respective positions of the main peak and the satellite peak on the theoretical pattern are fitted to the measured pattern first. The thicknesses and compositions of the metamorphic layers are adjusted in a recursive analysis until the calculated pattern agrees with the measured pattern, thereby providing an accurate analysis of laminated periodic structures.
However, when the data is over a large angular range and full of fringing then the density of local minima becomes very large. A typical local minimum condition is shown in FIG. 3. The process of trying to fit automatically the whole profile to the expected profile (FIG. 2) is rendered virtually impossible by the fact that some of the fringes overlap (when the profiles are overlaid) and give false minimum.
It is therefore an object of the invention to provide an improved method for determining parameters of a material.
According to the present invention, there is provided a method for determining parameters of a material, comprising
a) calculating an expected x-ray scattering profile for an approximation of said material,
b) obtaining an actual x-ray scattering profile of said material,
c) comparing a selected range of said expected x-ray scattering profile with a selected range of said actual x-ray scattering profile and generating error data based upon the differences between the profiles over the selected ranges,
d) modifying said expected x-ray scattering profile according to said error data,
e) repeating steps c) and d) until said range of the expected x-ray scattering profile substantially matches said range of the actual x-ray scattering profile,
f) expanding said range,
g) repeating steps c) to f) until said expected x-ray scattering profile substantially matches said actual x-ray scattering profile, and thereby determining parameters of the material from the parameters used for the last modified expected x-ray scattering profile.
Owing to the invention, it is possible to provide a method for determining parameters of a material that, by first fitting a range of the expected x-ray scattering profile to the actual x-ray scattering profile, reaches an accurate fit in a relatively short period of time, for even the more complicated x-ray scattering profiles.
Advantageously, prior to the method step c), a degree of smoothing is applied to the profiles. By applying smoothing to the profiles, it is easier and faster to match the two profiles. Preferably, the modifying of method step d) is achieved by the use of an iterative algorithm. It is also desirable, prior to the method step c) to first use a genetic algorithm to modify the expected x-ray scattering profile.
When selecting the range in the first run of the matching, it is advantageous that the range is the less sensitive region of the profile. The technique works especially well when the x-ray scattering profiles are x-ray reflectometry profiles and when the range includes the portion of the x-ray scattering profile where the scattered intensity is falling rapidly from the critical edge.